Ridler and Calvard’s, Kittler and Illingworth’s and Otsu’s methods for image thresholding
Introduction
In this correspondence, we aim to discuss the close relationships between three approaches to image thresholding, namely Ridler and Calvard (1978)’s or Trussell (1979)’s iterative-selection (IS) method, Kittler and Illingworth (1986)’s minimum-error-thresholding (MET) method and Otsu (1979)’s method.
With assumptions of bimodal or multimodal probability density functions of grey levels x, these three approaches are widely used in practice and highly cited by scientific publications. They are covered in some popular textbooks such as that written by Gonzalez and Woods, 2002, Gonzalez and Woods, 2008. The MET method is ranked as the best in a comprehensive survey of image-thresholding methods by Sezgin and Sankur (2004) recently. Otsu’s method is implemented as the default approach to image thresholding in some commercial and free software such as MATLAB (The MathWorks, Inc.) and GIMP (www.gimp.org).
The popularity of all three approaches is not a coincidence.
Recently, Xu et al. (2011) prove that, for image binarisation (or two-level thresholding), Otsu’s optimal threshold is the threshold t that equals the average value of the two class means, denoted by μ0(t) and μ1(t), for the two classes separated by t. That is, t = {μ0(t) + μ1(t)}/2. This result is in fact the iterative rule underlying the IS method. Such a link between the IS method and Otsu’s method has also been built by other studies, such as Reddi et al. (1984) and Magid et al. (1990).
Indeed, as we shall clarify more comprehensively in this correspondence, these three approaches are closely related to each other: briefly speaking, the IS method is an iterative version of Otsu’s method; Otsu’s method can be regarded as a special case of the MET method.
We shall show that, between the IS method, Otsu’s method and the MET method, the links can be readily built from the perspective of using a Gaussian-mixture distribution to model the grey-level distribution of an image, as indicated by Kurita et al. (1992) and Kittler and Illingworth (1986), among others. Such a perspective is different from, and complementary to, that of Reddi et al. (1984), Magid et al. (1990) and Xu et al. (2011).
In this context, although this correspondence may mainly revisit some results from various classical literature, our intentions are twofold. First, we intend to provide the practitioners with a more comprehensive clarification and some practical implications of the close relationships between these three popular approaches. Secondly, we intend to encourage further discussions about effectively applying, extending and evaluating the established image-thresholding approaches.
Section snippets
Relationships between the three approaches
Here we only consider image binarisation, but the discussions presented in the following sections can be readily generalised to multi-level thresholding.
Hence we assume that, in an image of N pixels, there are only two classes, and .
Let {p(x), x = 0, … , T}, for grey levels x, denote the normalised grey-level histogram constructed from the N pixels, such that and, by abuse of notation, we shall also use p(x) to denote the probability density function of x.
In addition, let y denote
Further Discussions
Our interpretations of the IS method, the MET method and Otsu’s method mainly follow that by Kittler and Illingworth (1986) and Kurita et al. (1992), based on statistical mixture models and the maximum likelihood estimation. There exist other interpretations of one or two of these methods from various perspectives, such as those in Kittler et al. (1985) based on simple statistics without using a histogram, in Yan (1996) based on a general weighted-cost function, in Morii (1991) and Jiulun and
Summary
In this correspondence, we have provided a comprehensive clarification of the close relationships between three popular image-thresholding approaches. That is, in short, Ridler and Calvard’s IS method is an iterative version of Otsu’s method; Otsu’s method can be regarded as a special case of Kittler and Illingworth’s MET method. It was our expectation that such a clarification could help the practitioners to understand more comprehensively the characteristics, thresholding performances and
Acknowledgements
The authors are grateful for the referees’ and the Area Editor’s constructive comments, in particular those on the unimodal thresholding and the evaluation of image-thresholding methods.
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